# Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations

arXiv: https://arxiv.org/abs/2106.01078 · HTML: https://mlsanity.com/paper/2106.01078/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 67 references that are in the map.
- First posted on arXiv 2 Jun 2021.

## Builds on (most cited first, 30 of 37)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids](https://mlsanity.com/paper/1810.01566/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Learning Compositional Koopman Operators for Model-Based Control](https://mlsanity.com/paper/1910.08264/) (2019)

## Built on by (most cited first, 6 of 6)

- [Knowledge-guided Machine Learning: Current Trends and Future Prospects](https://mlsanity.com/paper/2403.15989/) (2024)
- [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023)
- [Up-sampling-only and Adaptive Mesh-based GNN for Simulating Physical Systems](https://mlsanity.com/paper/2409.04740/) (2024)
- [Physics-Guided Learning of Meteorological Dynamics for Weather Downscaling and Forecasting](https://mlsanity.com/paper/2505.14555/) (2025)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)

## Abstract (arXiv)

Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from scientific data thrives as a new attempt to model complex phenomena in nature, but the effectiveness of current practice is typically limited by the scarcity of data and the complexity of phenomena. Especially, the discovery of PDEs with highly nonlinear coefficients from low-quality data remains largely under-addressed. To deal with this challenge, we propose a novel physics-guided learning method, which can not only encode observation knowledge such as initial and boundary conditions but also incorporate the basic physical principles and laws to guide the model optimization. We theoretically show that our proposed method strictly reduces the coefficient estimation error of existing baselines, and is also robust against noise. Extensive experiments show that the proposed method is more robust against data noise, and can reduce the estimation error by a large margin. Moreover, all the PDEs in the experiments are correctly discovered, and for the first time we are able to discover three-dimensional PDEs with highly nonlinear coefficients.